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Confined states and topological phases in two-dimensional quasicrystalline π-flux model

Rasoul Ghadimi, Masahiro Hori, Takanori Sugimoto, and Takami Tohyama
Phys. Rev. B 108, 125104 – Published 1 September 2023
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  • INTRODUCTION
  • AMMANN-BEENKER π-FLUX MODEL
  • PENROSE π-FLUX MODEL
  • SUMMARY AND DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Motivated by topological equivalence between an extended Haldane model and a chiral-π-flux model on a square lattice, we apply π-flux models to two-dimensional bipartite quasicrystals with rhombus tiles in order to investigate topological properties in aperiodic systems. Topologically trivial π-flux models in the Ammann-Beenker tiling lead to massively degenerate confined states whose energies and fractions differ from the zero-flux model. This is different from the π-flux models in the Penrose tiling, where confined states only appear at the center of the bands as is the case of a zero-flux model. Additionally, Dirac cones appear in a certain π-flux model of the Ammann-Beenker approximant, which remains even if the size of the approximant increases. Nontrivial topological states with nonzero Bott index are found when staggered tile-dependent hoppings are introduced in the π-flux models. This finding suggests a direction in realizing nontrivial topological states without a uniform magnetic field in aperiodic systems.

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    • Received 23 April 2023
    • Accepted 16 August 2023

    DOI:https://doi.org/10.1103/PhysRevB.108.125104

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Density of statesEdge statesElectronic structureFlat bandsHall effect
    1. Physical Systems
    QuasicrystalsTopological insulatorsTopological materials
    1. Techniques
    Exact diagonalization
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Rasoul Ghadimi1,2,3,*, Masahiro Hori4,*, Takanori Sugimoto5,6, and Takami Tohyama4

    • 1Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, South Korea
    • 2Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea
    • 3Center for Theoretical Physics (CTP), Seoul National University, Seoul 08826, South Korea
    • 4Department of Applied Physics, Tokyo University of Science, Tokyo 125-8585, Japan
    • 5Center for Quantum Information and Quantum Biology, Osaka University, Osaka 560-0043, Japan
    • 6Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan

    • *These authors contributed equally to this work.

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    Vol. 108, Iss. 12 — 15 September 2023

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    Images

    • Figure 1
      Figure 1

      (a) Ammann-Beenker tiling. The indicated area with green dashed edges refers to a symmetric domain of tiles called D1 that is used in Table 1. (b) An example of Ammann-Beenker tiling approximant (g=1). The numbers inside circles show approximant sublattice indices. Blue dots show the origin of a unit cell and blue arrows represent the basis vectors of the unit cell, a⃗1, a⃗2. (c) Energy levels (black points) sorted from lowest to highest energies vs index of levels, 1 to Nt in the bottom horizontal axis and DOS (blue lines) for a zero-flux model in an Ammann-Beenker approximant [Nt=2×2×1393]. (d) Edges of the Ammann-Beenker tiling drawn by arrows starting from a vertex of the set A. The colored arrows correspond to those in (e). (e) Phase An (=π/2, n=1,⋯,4) to construct quasicrystalline π-flux models. The direction of arrows corresponds to R̂n (see the text).

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    • Figure 2
      Figure 2

      [(a1),(b1),(c1)] Different π-flux configurations named AB.conf1–3 for the Ammann-Beenker tiling, where red, blue, and white regions indicate tiles with −π, +π, and zero flux, respectively. Arrows indicate the presence of nonzero phases on the edge of tiles. Energy levels and DOS for (a2) AB.conf1, (b2) AB.conf2, and (c2) AB.conf3. [(a3),(b3),(c3)] Same as (a2), (b2), and (c2) but STDH with td=0.3t is used. The energy levels as a function of STDH td for (a4) AB.conf1, (b4) AB.conf2, and (c4) AB.conf3. [(a5),(b5),(c5)] Same as (a4), (b4), and (c4) but NSTDH is used. Edge modes at a given energy E in the case of STDH td=0.3t for (a6) AB.conf1, (b6) AB.conf2, and (c6) AB.conf3, where the red (blue) region indicates a larger (smaller) probability of an edge-mode wave function.

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    • Figure 3
      Figure 3

      (a) Energy spectrum with OBC (red lines) and PBC (blue lines) for AB.conf2 as a function of STDH td. (b) The distribution of a wave function with energy E≈1.1t with OBC for different td. In (a) and (b) we use the same system as we considered in Fig. 2. [(d)–(f)] Band dispersion of a π-flux AB.conf2 model along high-symmetry line for td=0 (blue) and td=0.1t (red) with different approximant sizes. (d) g=1, (e) g=2, and (f) g=3. In the inset of (f) we plot Brillouin zone (BZ) and indicate k⃗Γ=0, k⃗M=12b⃗1+12b⃗2, and k⃗K=12b⃗1.

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    • Figure 4
      Figure 4

      Penrose π-flux model. (a) Energy spectrum (black points) and DOS (blue lines) for approximant of Penrose tiling without flux with g=8 [Nt=9349]. (b) Penrose tiling where arrows are attached. (c) An for the Penrose tiling. [(d1),(e1),(f1)] Possible Penrose π-flux models. [(d2),(e2),(f2)] Corresponding energy spectra as a function of STDH td, where red (green) dots indicate topologically nontrivial (trivial) gaps.

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    • Figure 5
      Figure 5

      The π-flux model in the square lattice. Two tiles of (a) π flux, and π flux with (b) staggered tile diagonal hopping (STDH), and (c) nonstaggered tile diagonal hopping (NSTDH). The two green symbols inside the tiles indicate positive and negative π [π/2] flux in (a) [(b) and (c)]. In (c), red and blue diagonal links indicate hopping with opposite signs regarding each other. (d) The π-flux model in the square lattice, where a1⃗,a2⃗ show basis vectors. The energy dispersion for (e) td=0, (f) STDH with td=0.1t, and (g) NSTDH with td=0.1t along high-symmetry lines.

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    • Figure 6
      Figure 6

      (a) Arrow-attached Ammann-Beenker and (c) arrow-attached Penrose tilings. (b) Inflation rule for the Ammann-Beenker tiling and (d) that for the Penrose tiling.

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    • Figure 7
      Figure 7

      Penrose tiling, where the bipartite index of vertices is given by red and blue dots. The green lines show the forbidden ladder and separate regions with different bipartite imbalances.

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    • Figure 8
      Figure 8

      Cluster-1 of Pe.conf1 for the π-flux model. The angle of the cluster is different between (a) and (b). The arrows denote the hopping with nonzero Peierls phase (see the main text for definition).

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    • Figure 9
      Figure 9

      Six types of confined states in the zero-flux model on Penrose tiling. The number on each vertex represents the relative components of the wave function.

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    • Figure 10
      Figure 10

      Six types of the tiles with r=0 on Penrose tiling.

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    • Figure 11
      Figure 11

      The type-3 tiles of Pe.conf1in the π-flux model. (a) r=0 and (b) r=3. The arrows denote the hopping with nonzero Peierls phase (see the main text for definition).

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    • Figure 12
      Figure 12

      Some examples of the confined states in Pe.conf1 [(a)–(d)], Pe.conf2 [(e)–(h)], and Pe.conf3 [(i)–(l)] of the π-flux model on Penrose tiling. The tiles are (a) type-2 with r=1, (b) type-3 with r=3, (c) type-4 with r=3, (d) type-5 with r=1, (e) type-2 with r=0, (f) type-3 with r=4, (g) type-4 with r=4, (h) type-5 with r=0, (i) type-2 with r=1, (j) type-3 with r=3, (k) type-4 with r=3, and (l) type-5 of r=1. Each number written on the vertices is the relative component of the wave functions for the confined states with zero energy. The arrows denote the hopping with nonzero Peierls phase (see the main text for definition).

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    • Figure 13
      Figure 13

      Ammann-Beenker tiling. The red, blue, and green shaded areas are D1, D2, and D3, respectively. The sites on the boundary that are indicated by blue dashed lines are not included in the domains. For example, the red, blue, and green dots at the boundary of D1, D2, and D3 are excluded.

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    • Figure 14
      Figure 14

      The confined state in D1 for AB.conf1 of the π-flux model with E=2 on Ammann-Beenker tiling. The number represents the wave function's component of the confined state on each vertex. The arrows denote the hopping with nonzero Peierls phase (see the main text for definition).

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    • Figure 15
      Figure 15

      (a) Schematically drawn Hamiltonian of π-flux model for i=3 Ammann-Beenker tiling. The hopping value from vertex i to j is 1, ı, and −ı black lines connecting i and j, green arrows from i to j, and green arrows from j to i, respectively. (b) The area is represented by the red rectangle in (a). The yellow-shaded regions are D1 structures.

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    • Figure 16
      Figure 16

      τs2i (black line) and Ninet (blue circles) as a function of generation i. Ninet is fitted by aτsbi shown by the red line.

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    • Figure 17
      Figure 17

      The generation i dependence of p(i). p(i) is plotted by blue circles, while the fitted result using c(1−τsdi) is shown by the red line.

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    • Figure 18
      Figure 18

      The confined state in D1 for AB.conf2 in the π-flux model on Ammann-Beenker tiling with E=0. Each number on the vertices represents the relative wave function's components of the confined state at each vertex.

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    • Figure 19
      Figure 19

      The confined state in D2 for the π-flux model of AB.conf2 with E=0 on Ammann-Beenker tiling. Each number on the vertices represents the amplitude of the confined state at each site. The arrows denote the hopping with nonzero Peierls phase (see the main text for definition).

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    • Figure 20
      Figure 20

      Two confined states, Ψ12,2 and Ψ22,2, for AB.conf2 in the π-flux model on Ammann-Beenker tiling with E=2. Each number on the vertices represents the wave function's components of the confined state at each vertex. The arrows denote the hopping with nonzero Peierls phase (see the main text for definition).

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    • Figure 21
      Figure 21

      Seven confined states, {Ψk3,0}k(1≤k≤7), for AB.conf3 in the π-flux model on Ammann-Beenker tiling with E=0. Utilizing the even or odd mirror symmetry of confined states, one-fourth of the entire tiling of Ψ63,0 and Ψ73,0 are shown. Each number on the vertices represents the amplitude of the confined state at each vertex. The arrows denote the hopping with nonzero Peierls phase (see the main text for definition).

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    • Figure 22
      Figure 22

      The confined state Ψ13,22 for AB.conf3 in the π-flux model on Ammann-Beenker tiling with E=22. Each number on the vertices represents the amplitude of the confined state at each vertex. The arrows denote the hopping with nonzero Peierls phase (see the main text for definition).

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    • Figure 23
      Figure 23

      The generation i dependence of p(i) for the π-flux model on Ammann-Beenker tiling. “Conf.c, E=e” corresponds to the configuration c with energy e, where configuration 0 is the zero-flux model. The converged value of limi→∞p(i), which is obtained by calculating p in Eq.(F3) except for the AB.conf2 with E=0, is plotted at i=∞. The converged value of limi→∞p(i) for the AB.conf2 with E=0 is obtained by the fitting in Fig. 17.

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